MathDB

1

Prove reflections of a point over sides of a parallelogram are concyclic

P

be a point in parallelogram

ABCD

such that

PA \cdot PC + PB \cdot PD = AB \cdot BC.

Prove that the reflections of

P

over lines

AB

,

BC

,

CD

, and

DA

are concyclic.

1

Finding the maximum number of two-way trips

Let

n

and

\ell

be integers such that

n \ge 3

and

1 < \ell < n

. A country has

n

cities. Between any two cities

A

and

B

, either there is no flight from

A

to

B

and also none from

B

to

A

, or there is a unique two-way trip between them. A two-way trip is a flight from

A

to

B

and a flight from

B

to

A

. There exist two cities such that the least possible number of flights required to travel from one of them to the other is

\ell

. Find the maximum number of two-way trips among the

n

cities.

6

1

BAD CAT geometry

Let

D

be an interior point of triangle

ABC

. Lines

BD

and

CD

intersect sides

AC

and

AB

at points

E

and

F

, respectively. Points

X

and

Y

are on the plane such that

BFEX

and

CEFY

are parallelograms. Suppose lines

EY

and

FX

intersect at a point

T

inside triangle

ABC

. Prove that points

B

,

C

,

E

, and

F

are concyclic if and only if

\angle BAD = \angle CAT

.

3

Infinite positive integer sequence with an eventual cubic lower bound

Let

a_1, a_2, a_3,\ldots

be an infinite sequence of positive integers such that

a_2 \ne 2a_1

, and for all positive integers

m

and

n

, the sum

m + n

is a divisor of

a_m + a_n

. Prove that there exists an integer

M

such that for all

n > M

, we have

a_n \ge n^3

.

Find all triples such that a quadratic bound implies a constant function

Determine all ordered triples

(a, b, c)

of real numbers such that whenever a function

f : \mathbb{R} \to \mathbb{R}

satisfies

|f(x) - f(y)| \le a(x - y)^2 + b(x - y) + c

for all real numbers

x

and

y

, then

f

must be a constant function.

Circle with diameter AC - AB

Given

\triangle ABC

with

AB < AC

, let

\omega

be the circle centered at the midpoint

M

of

BC

with diameter

AC - AB

. The internal bisector of

\angle BAC

intersects

\omega

at distinct points

X

and

Y

. Let

T

be the point on the plane such that

TX

and

TY

are tangent to

\omega

. Prove that

AT

is perpendicular to

BC

.

2

11 labelled points and a harmonic length condition

In a triangle

ABC

with circumcircle

\Gamma

,

M

is the midpoint of

BC

and point

D

lies on segment

MC

. Point

G

lies on ray

\overrightarrow{BC}

past

C

such that

\frac{BC}{DC} = \frac{BG}{GC}

, and let

N

be the midpoint of

DG

. The points

P

,

S

, and

T

are defined as follows: [list = i] [*] Line

CA

meets the circumcircle

\Gamma_1

of triangle

AGD

again at point

P

. [*] Line

PM

meets

\Gamma_1

again at

S

. [*] Line

PN

meets the line through

A

that is parallel to

BC

at

Q

. Line

CQ

meets

\Gamma

again at

T

.
Prove that the points

P

,

S

,

T

, and

C

are concyclic.

Degree 2019 functional equation

Find all functions

f : \mathbb{R} \to \mathbb{R}

that satisfy the equation

f(x^{2019} + y^{2019}) = x(f(x))^{2018} + y(f(y))^{2018}

for all real numbers

x

and

y

.

2019 Philippine TST

Subcontests

Prove reflections of a point over sides of a parallelogram are concyclic

Finding the maximum number of two-way trips

BAD CAT geometry

Infinite positive integer sequence with an eventual cubic lower bound

Find all triples such that a quadratic bound implies a constant function

Circle with diameter AC - AB

11 labelled points and a harmonic length condition

Degree 2019 functional equation